INTERMED MICRO ECON 300
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Date Created: 09/09/15
Econ 495 1 Describing Risk To bring our models closerto reality we would like to be able to describe decisions that are made in an uncertain world A full description of uncertain outcomes would involve an itemization of all possible outcomes together with a description ofthe probability of each outcome The probability of an outcome is the likelihood of that outcome EX p1X1 p2X2 pn Xn For example suppose that M T Brain is considering two and only two jobs Job A as comptroller for a school offers him a sure salary of 30000 Job B as comptroller for Red Cap a software company will pay him 50000 in salary and bonus during good times 50 ofthe time but only 10000 during pad times 50 of the time What is the expected value ofthe uncertain outcome Write the probability of each outcome as pi Then the expected value ofworking for Red Cap is EX p110000 p250000 0510000 0550000 500025000 30000 Which of the two jobs will Mr Brain choose Although the expected value ofthe two jobs is identical we need to know how Mr Brain views the variance of an uncertain outcome Variance is the expected value ofthe squared deviations between the possible outcomes and the expected value of the average payoff How do you calculate the variance 1 First calculate the expected value ofthe choice 30000 for expected income at Red Cap 2 Then calculate the difference between each possible outcome and the average expected value ofthat choice and square that result For Red Cap calculate 1000030000 20000 and 5000030000 20000 Squaring each yields 400000000 Squaring yields a positive value to assure that positive and negative deviations don t cancel For each possible outcome multiply the squared deviation by the associated probability and add up all the outcomes The result is the variance For Red Cap this is 05400000000 05400000000 400000000 Once we have the variance we can take the square root ofthat number called the standard deviation and that square root will always be positive 4 The standard deviation is simply the square root ofthe variance or 20000 A 2 Risk preferences How would Mr Brain evaluate each of these alternatives We would like to be able to compare his utility from each ofthe choices Suppose that Mr Brain s utility function with respect to income has the form UI J7 That is we assume that greater income increases his utility but at a decreasing rate Thus Mr Brain s expected utility from a guaranteed income of 30000 is U30000 J7 1732 A person with a diminishing marginal utility of income would be risk averse In fact most people are risk averse They buy life insurance and health insurance and they require a higher wage for a risky job How do we evaluate the value of an uncertain outcome Let us de ne expected utility as the expected value ofthe utility levels that Mr Brain would receive from each payoff Thus Mr Brain s expected utility of working for Red Cap is EVU p1u10000 p2 u 5000005JZ05JZ 5100 52236 5011181618 So Mr Brain receives a lower expected utility from the uncertain outcome with an expected value of 30000 How much lower is the guaranteed wage that would give him the same utility level as the uncertain outcome lfhe had a sure income of 2617924 or 161 82 then he would have the same utility level as he has with the uncertain outcome His risk premium is about 3821 u 5mm u u mm 31 52 um 5mm u mum 2mm auuun mum suuuu snuuu v 47000 21679 211 48000 21909 214 49000 22136 217 50000 22361 220 51000 22583 223 3 Since a riskaverse person is willing to sacri ce some income in order to reduce risk we would expect him to be willing to purchase insurance against the consequences of bad states of the world Suppose that an investor is building a bottling plant in a foreign country If all goes well the investor will have an income of100000 at the end ofthe year However if the local government expropriates your plant you will have 0 Let s suppose that the probability of expropriation is 02 so that the probability of not being expropriated is 08 You have the opportunity to buy 100000 worth of insurance for 20000 Is this insurance fairly priced Does the insurance payment equal the expected value ofthe promised payment The expected value is EV 02 100000 080 20000 Yes it is fairly priced because the insurance premium per dollar of insurance coverage 02 equals the probability of expropriation Should you buy this insurance Yes the expected value of your income is the same and the insurance allows you to eliminate all of your risk 4 Would you expect a fully insured person to choose their projects with the same care as an uninsured person lfthey thought that their current choices would not affect their future insurance rates then there might be opportunities for a Moral hazard The individual might exercise less care in choosing investments if he were fully insured against all losses Adverse selection lfthe availability of insurance attracted individuals who knew that the underlying risk of their projects was actually greater than 02 then the insurer would end up attracting the riskiest customers 039 5 Decision Analysis Decisionmaking under uncertainty is easily described by laying out the choices and the uncertain outcomes in a decision tree I n the decisiontree below choice nodes are indicated by a rectangle and an uncertain probability indicated by a circle Beginning at the end of the problem with the value of the payoffs a riskneutral decisionmaker can bring the uncertain payoffs back to calculate the expected value of an uncertain outcome viewed before thefact Large POO39 100 mil Build large Rig 10 m Small pool Large pool 40 mil Build small Rig 25 mil Sma POO39 Build large Rig Explora ry drilling 100 mil 10 mil Cost 10 mil Test large 25 mil 10 mil Build small Rig Consider the problem faced by Shell Oil They can build a large offshore rig on Sakhalin or they can build a small onshore facility which costs the same amount but accesses the reserve through slantdrilling Before the fact they don t know the size ofthe oil pool they wish to drill Their seismic exploration indicates that there is a 25 percent chance the pool is large and a 75 percent chance that it is small Their payoffs conditional on the drilling technique they choose and the size ofthe pool are indicated at the far right of the diagram What should they do lfthey drill without conducting any exploratory drilling then the expected value of building a large rig is EVL 25 100 751O 325 mil Their expected value of building a small rig is EVS 254O 7525 2875 mil Alternatively they can undertake exploratory drilling which will cost 10 mil but it will provide a 100 percent accurate forecast of the size ofthe pool Before the fact there is a 25 percent chance the exploration will nd a large pool and a 25 percent chance it will nd a small pool lfthey nd a large pool they will invest in a large rig lfthey nd a small pool they will invest in a small rig The expected value of exploratory drilling is EVED 25 10010 75 2510 3375 80 they should undertake exploratory drilling Note that the value of the information they get from exploratory drilling is the difference between 4375 mil 251007525 and 325 million Or the value ofthe information is 1125 million and the cost of getting the information is 10 million
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